On uniqueness in electromagnetic scattering from biperiodic structures
ESAIM: Mathematical Modelling and Numerical Analysis , Direct and inverse modeling of the cardiovascular and respiratory systems. Numéro spécial, Tome 47 (2013) no. 4, pp. 1167-1184.

Consider time-harmonic electromagnetic wave scattering from a biperiodic dielectric structure mounted on a perfectly conducting plate in three dimensions. Given that uniqueness of solution holds, existence of solution follows from a well-known Fredholm framework for the variational formulation of the problem in a suitable Sobolev space. In this paper, we derive a Rellich identity for a solution to this variational problem under suitable smoothness conditions on the material parameter. Under additional non-trapping assumptions on the material parameter, this identity allows us to establish uniqueness of solution for all positive wave numbers.

DOI : 10.1051/m2an/2012063
Classification : 35A02
Mots-clés : biperiodic scattering, uniqueness, electromagnetic waves 
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Lechleiter, Armin; Nguyen, Dinh-Liem. On uniqueness in electromagnetic scattering from biperiodic structures. ESAIM: Mathematical Modelling and Numerical Analysis , Direct and inverse modeling of the cardiovascular and respiratory systems. Numéro spécial, Tome 47 (2013) no. 4, pp. 1167-1184. doi : 10.1051/m2an/2012063. http://www.numdam.org/articles/10.1051/m2an/2012063/

[1] T. Abboud, Electromagnetic waves in periodic media, in Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, Newark, DE. SIAM, Philadelphia (1993) 1-9. | MR | Zbl

[2] H. Alber, A quasi-periodic boundary value problem for the laplacian and the continuation of its resolvent. Proc. Royal Soc. Edinburgh 82 (1979) 251-272. | MR | Zbl

[3] T. Arens, Scattering by biperiodic layered media: The integral equation approach.Habilitation Thesis, Universität Karlsruhe (2010).

[4] G. Bao, Variational approximation of Maxwell's equations in biperiodic structures. SIAM J. Appl. Math. 57 (1997) 364-381. | MR | Zbl

[5] G. Bao, L. Cowsar and W. Masters, Mathematical modeling in optical science. SIAM Frontiers Appl. Math. SIAM, Philadelphia (2001). | MR | Zbl

[6] G. Bao and D.C. Dobson, On the scattering by a biperiodic structure. Proc. Amer. Math. Soc. 128 (2000) 2715-2723. | MR | Zbl

[7] A.-S. Bonnet-Bendhia and F. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Methods Appl. Sci. 17 (1994) 305-338. | MR | Zbl

[8] S.N. Chandler-Wilde and P. Monk, Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces. SIAM. J. Math. Anal. 37 (2005) 598-618. | MR | Zbl

[9] M. Costabel, M. Dauge and S. Nicaise, Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains. http://hal.archives-ouvertes.fr/hal-00453934/.

[10] D. Dobson and A. Friedman, The time-harmonic Maxwell's equations in a doubly periodic structure. J. Math. Anal. Appl. 166 (1992) 507-528. | MR | Zbl

[11] D.C. Dobson, A variational method for electromagnetic diffraction in biperiodic structures. Math. Model. Numer. Anal. 28 (1994) 419-439. | Numdam | MR | Zbl

[12] H. Haddar and A. Lechleiter, Electromagnetic wave scattering from rough penetrable layers. SIAM J. Math. Anal. 43 (2011) 2418-2433. | MR | Zbl

[13] W. Mclean, Strongly Elliptic Systems and Boundary Integral Operators. Cambridge University Press, Cambridge, UK (2000). | MR | Zbl

[14] P. Monk, Finite Element Methods for Maxwell's Equations. Oxford Science Publications, Oxford (2003). | Zbl

[15] F. Rellich, Darstellung der Eigenwerte von Δu + λu = 0 durch ein Randintegral. Math. Zeitschrift 46 (1940) 635-636. Doi: 10.1007/BF01181459. | JFM | MR | Zbl

[16] G. Schmidt, On the diffraction by biperiodic anisotropic structures. Appl. Anal. 82 (2003) 75-92. | MR | Zbl

[17] C. Wilcox, Scattering Theory for Diffraction Gratings. Appl. Math. Sci. Springer-Verlag 46 (1984). | MR | Zbl

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